Page 306 - Modelling in Transport Phenomena A Conceptual Approach

286 CHAPTER 8. STEADY MlCROSCOPIC BALANCES WITHOUT GEN.

8.4.1 Diffusion in Rectangular Coordinates
Consider the transfer of species A by diffusion through a slightly tapered slab as
shown in Figure 8.26.

Figure 8.26 Diffusion through a slightly tapered conical duct.

If the taper angle is small, mass transport can be considered one-dimensional in
the z-direction. Since XA = XA(Z), Table C.7 in Appendix C indicates that the
only non-zero molar flux component is NA, and it is given by

NA, = J;, = -CDAB - (8.47)
dXA
dz
Note that the negative sign in Eq. (8.47) implies that positive z-direction is in
the direction of decreasing concentration. If the answer turns out to be negative,
this implies that the flux is in the negative x-direction.
Over a differential volume element of thickness Az, as shown in Figure 8.26,
Eq. (8.41) is written as
(ANA.)I, - (ANA,)I.+A, =O (8.48)

Dividing Eq. (8.48) by Az and taking the limit as Az + 0 gives

(8.49)

(8.410)
Since flux times area gives the molar transfer rate of species A, ~ZA, it is possible
to conclude that
ANA, = constant = ?LA (8.411)
in which the area A is perpendicular to the direction of mass flux.
Substitution of Eq. (8.47) into Eq. (8.411) and integration gives

(8.412)